Speaker
Description
Light-matter interactions in the strong-field regime, such as high-harmonic generation (HHG), typically give rise to highly-oscillatory
integrals, which are often solved using saddle-point methods. Not only do these methods promise a faster computation, but they also inform a more intuitive understanding of the process in terms of quantum orbits, as the saddle points correspond to interfering quantum trajectories (i.e., Feynman’s path integral formalism).
Despite these advantages, a sound understanding of how to rigorously apply saddle-point methods to highly-oscillatory integrals in high dimensionality with algorithms that work uniformly for arbitrary configurations and laser fields, remains conspicuously absent. This hinders our ability to keep up with state-of-the-art experimental setups which increasingly rely on tightly-controlled laser waveforms.
We introduce the key ideas of Picard-Lefschetz theory – the foundation of all saddle-point methods – and their implementation. Using HHG and above-threshold ionisation driven by two-colour laser fields as examples, we show how those ideas provide a universal and robust approach for a fast computation of the integrals, as well as a widely-applicable algorithm to derive the relevant semi-classical quantum orbits that underlie the physical processes. We demonstrate this for setups that involve spectral caustics, as well as intensity scans for a two-colour field beyond the perturbative regime (“colour switchover”), which were previously rendered impossible to solve with saddle-point methods due to coalescences and branch cuts.